In order the limit to exist, should be true. But doesn't exist because of that square root. Some books assume that limits should be evaluated as one-sided limits at the endpoints. In my opinion, the question should've been to avoid confusion.

Awesome! But isn't this kind of using Cauchy's Mean Value Theorem...which is basically L'hopital's?

Isn't it sort of a Taylor series expansion ? I wasn't very clear with "approximations" stuff... But once again feel free to use the method you like (I just point you towards one in particular lol). The only rule is no l'hospital's ^^